I'm wondering if you had some other event that impacted your clients after performing a license manager update. If your clients weren't able to access their license, you should seen an error message stating that the license was unavailable, and shouldn't prevent the ArcFM login screen from appearing. Would you be able to check what license is selected in your 'Desktop Administrator' for the client you are attempting to run ArcFM from? You should see something along the lines of the following:
Your product may vary, but it should not be set to <No Product>. That's the only way I am aware of to disable the ArcFM splash screen on a client after installing the ArcFM software.